`log_10(t - 3) = 2.6` Solve the equation accurate to three decimal places

Monday, August 22, 2011

In solving a logarithmic equation, we may simplify using logarithm properties.

Recall the logarithm property: $\displaystyle{{a}}^{{{\left({\log}_{{{a}}}{\left({x}\right)}\right)}}}={x}$ .

When we raise the log with the same base, the "log" will cancel out.

This is what we need to accomplish on the left side.

The problem $\displaystyle{\log}_{{{10}}}{\left({t}-{3}\right)}={2.6}$ has a logarithmic base of 10.

That is our clue to raise both sides by base of 10.

$\displaystyle{{10}}^{{{\log}_{{10}}{\left({t}-{3}\right)}}}={{10}}^{{{2.6}}}$

$\displaystyle{t}-{3}={{10}}^{{{2.6}}}$

$\displaystyle+{3}+{3}$

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$\displaystyle{t}={{10}}^{{{2.6}}}+{3}$

$\displaystyle{t}\approx{401.107}$

To check, plug-in the value of $\displaystyle{t}={401.107}$ in $\displaystyle{\log}_{{{10}}}{\left({t}-{3}\right)}$ :

$\displaystyle{\log}_{{{10}}}{\left({401.107}-{3}\right)}$

$\displaystyle{\log}_{{{10}}}{\left({398.107}\right)}$

$\displaystyle={2.599999814}\approx{2.6}$

So,

$\displaystyle{t}\approx{401.107}$ is a real solution.

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